Unfortunately, these two things have absolutely no relationship to each other.
Recently, while working on a sheet metal project, I was annoyed that I had to go find a sheet metal gauge chart, because sheet metal is pretty much only sold in “gauge” thickness. I was even more annoyed to find that different metals have different gauges. So 14 gauge stainless is not the same as 14 gauge copper, and neither of these are the same as 14 gauge aluminum
Although this is an outdated system, its only modern vestige applies to shotguns.
And it goes like this:
The bore gauge number (n) is the diameter of the bore, such that n lead balls of equal diameter weigh 1 pound.
Therefore, the greater the bore gauge, the smaller the diameter.
So when is 12 > 20? When you’re talking about bore gauge.
Yep. That’s bad. But, it’s at least manageably bad. There is at least a formula.
Lead weighs about 11.8 ounces of weight per ounce of volume.
A pound of lead (16 weight ounces) would therefore occupy a mere 16 / 11.8 = 1.35 fluid ounces.
To relate volume and diameter, we have to convert fluid ounces to inches, and 1 ounce of volume = 1.8 cubic inches.
Therefore, 1.35 fluid ounces of lead = 1.35 * 1.8 = 2.43 cubic inches.
At this point we have 2.43 cubic inches of lead, which we know is enough for a “1 gauge” lead ball, but we need to find the diameter.
Volume of a sphere (V) = 4 / 3 * Pi * r^3
r = Radius
r^3 = Radius to the 3rd power (cubed)
Since we have the volume and we need the diameter, we need to rearrange this to solve for radius, then multiply by 2 (diameter = 2 * radius)
d = 2 * ( (3 * V) / (4 * Pi)) ^ 0.3333
By inverting ^3 (cubed) as ^ 0.3333 we get the cubed root
If we compute this for our 2.43 cubic inches of lead, we get:
d = 2 * ( (3 * 2.43) / (4 * Pi)) ^ 0.3333
d = 2 * ( 7.29 / 12.57 ) ^ 0.3333
d = 2 * 0.58 ^ 0.3333
d = 2 * 0.83
d = 1.667 inches
A 1 gauge bore diameter is 1.667 inches.
For the general case, we can start by computing the amount of lead. Since volume and weight are proportional, and the volume of a pound of lead is constant:
V = 2.43 / g
2.43 = 1 pound of lead in cubic inches
g = gauge number
We then take the same formula, substitute for V, and pre-compute some constants:
d = 2 * ( ( 3 * 2.43 / g ) / (4 * Pi )) ^ 0.3333
d = 2 * ( 0.58 / g ) ^ 0.3333
We can cube both sides and further simplify:
d ^ 3 = 8 * 0.58 / g
d ^ 3 = 4.64 / g
d = 1.668 / g ^ 0.3333
And, we can use this for any gauge number (g):
||d = 1.668 / 10 ^ 0.3333
d = 1.668 / 2.154
d = 0.77 inches
||d = 1.668 / 12 ^ 0.3333
d = 1.668 / 2.289
d = 0.73 inches
||d = 1.668 / 20 ^ 0.3333
d = 1.668 / 2.714
d = 0.61 inches
And, you can plug any number in to this formula, so if you wanted a 14.5 gauge bore size (for whatever reason), you would use the same formula above, and that bore size is 0.68 inches.
You can even reverse the process to find the bore gauge corresponding to a given diameter measured in inches, by solving for gauge:
d = 1.668 / g ^ 0.3333
d ^ 3 = 4.64 / g
g = 4.64 / d ^ 3
A .357 magnum has a nominal bore diameter of 0.354 inches. If we plug in 0.354, we get:
g = 4.64 / 0.354 ^ 3
g = 4.64 / 0.0444
g = 104.6
So the next time you threaten someone, you can say “Don’t make me pull out my 104 gauge!”, and at least this has an obscure but discernible meaning.
How did we get stuck with this standard? Nobody really knows! However, most texts state that gauge was used even as late as 200 years ago, when there weren’t any good, universal measuring standards. If you ordered ammunition in gauge size, you were assured that it would match your gun’s bore diameter. As the rifle (gun with a rifled barrel) is a relatively new invention, measurements were much more standardized by that time, and it makes sense that rifle bore sizes (caliber) are always measured in inches or millimeters.
However, to this day, most shotguns are still smooth-bore, because shotgun cartridges have multiple, small projectiles (called pellets) that spray out in a disc formation when the cartridge is fired. Because the pellets are much smaller than the shotgun’s bore, they don’t come in to contact with the barrel enough for rifling to be effective. Although there are rifled shotguns (called slug guns) that are designed for shooting a single, solid lead shot (called a slug), these are uncommon. Therefore, it makes sense that we’ve kept the tradition of measuring a shotgun’s smooth bore size in gauge, rather than using caliber.
Fortunately, today, bore gauge is only used for shotguns, and rifle bores have become so standardized that I can go anywhere in the world and purchase ammunition for a gun of the same caliber manufactured anywhere else in the world.
Now, let’s talk about sheet metal gauge.
Sheet Metal Gauge
Here is the formula for sheet metal gauge:
There isn’t one. Consult a gauge chart.
But, it gets even better.
Sheet metal gauge is based on density, so not only do you need a gauge chart, you need a different gauge chart for each type of sheet metal.
How did we end up with this?
Sheet metal gauge size is based on wire gauge size, but not AWG, which is the American standard for wire gauge size.
Back in the early days, each vendor had their own wire gauge size, based on the number of machining steps required in order to get down to that gauge size. E.g. 10 gauge wire requires 10 steps. Eventually, all of that got standardized as the American Wire Gauge (AWG) standard.
Wire has been mass-produced much longer than sheet metal, so when sheet metal started to be mass-produced, the sheet metal manufacturers adopted the wire gauge standard (but not AWG), and each manufacturer had their own.
Over time, like AWG, sheet metal gauge sizes became standardized.
Unlike AWG which specifies a standard wire diameter independent of any other factor (e.g. 20 gauge wire is the same diameter in copper or aluminum, or steel, or whatever), sheet metal gauge is based on density, so the gauge size changes based on the specified material.
Nominally, every reference to sheet metal states that gauge size is based on “a density of 41.82 pounds per square foot”, but that’s not the case. If it was the case, you would be able to work backwards to find the gauge number. For example, going back to the way bore gauge works, if 41.82 pounds = 1 (sheet metal) gauge, then dividing that by 2 = 2 gauge, etc. But that’s not the case. 10 gauge (steel) = 0.1345 inches, and if we multiply that by 10, we get 1.35 inches. 20 gauge = 0.0359 inches, and if we multiply that by 20, we only get 0.72 inches. So there is clearly no proportionality based on weight or thickness.
If you google for a sheet metal gauge formula, you can find plenty of references that all begin with:
Sheet metal gauge size is based on 41.82 pounds per square foot. Consult your sheet metal gauge chart, and…
As it turns out, this is a formula for calculating the weight based on gauge size, not for calculating gauge size.
Why do we still use this archaic standard? Sheet metal gauge is codified by U.S. law, for tax purposes: 15 USC 206: Standard gauge for sheet and plate iron and steel.
So, because the government regulates the thickness of sheet metal for tax purposes, you have to consult a gauge chart whenever you order sheet metal.